Question 140·Medium·Systems of Two Linear Equations in Two Variables
In the solution to the system of equations above, what is the value of ?
(Express the answer as an integer)
For systems of two linear equations, first check if elimination is easy—look for coefficients of a variable that are already opposites or can quickly be made opposites. Add or subtract the equations to eliminate one variable, solve for the remaining variable, then substitute back into either original equation to find the second variable. Keep arithmetic neat and simple, and quickly verify your solution by plugging both values into the other equation to see if it holds true.
Hints
Choose a method
You can use elimination or substitution. Look at the two equations and see if one variable can be canceled easily by adding or subtracting the equations.
Look at the y-terms
The first equation has and the second has . What happens to the terms if you add the two equations together?
After finding x, don’t stop
Once you solve for , plug that value into one of the original equations to solve for .
Desmos Guide
Enter the equations
Type y = 13 - 2x as the first line in Desmos (this is the same as solved for ).
Enter the second equation
Type y = 3x - 17 as the second line in Desmos (this is the same as solved for ).
Find the intersection point
Look at where the two lines intersect on the graph. Click the intersection point to see its coordinates and read off the y-coordinate; that is the value of in the solution to the system.
Step-by-step Explanation
Eliminate one variable by combining the equations
Notice that the first equation has and the second has :
If you add the two equations, the terms will cancel:
The left side simplifies to and the right side to , so you get an equation in terms of only.
Solve for x
From Step 1, you have:
Divide both sides by 5:
Now you know the -value in the solution .
Substitute x back to find y
Substitute into either original equation. Using :
Subtract 12 from both sides:
So, in the solution to the system, the value of is 1.